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% name as ATO: Lissajous Curves, e.g. Knot 7_4.pdf

\title{\phantom{.} \vskip-3cm
\huge Lissajous Curves, e.g. the Prime Knot $7_4$
 \footnote{\large This file is from the 3D-XploreMath project. 
\hfil\break Please see http://www.math.uci.edu/$\sim$palais/  or http://3d-xplormath.org/}}
\author{}
\begin{document}

\maketitle
\vskip-2cm
\LARGE
Lissajous curves are a popular family of planar curves, resp. space curves. They are complicated
enough to be interesting, but regular enough to be esthetically pleasing. They are described
by simple formulas:
$$ x(t) := aa \cdot \sin(2\pi\cdot dd  \cdot  t) \phantom{+ gg}$$
$$ y(t) := bb \cdot \sin(2\pi\cdot ee  \cdot  t + gg) $$
$$ z(t) := aa \cdot \sin(2\pi\cdot ff  \cdot  t + cc) $$
In 3DXM the parameters $dd, ee, ff$ are rounded to integers so that the curves are closed on
the interval $[0,\pi]$. The default morph varies the phase $gg$ from $0$ to $\pi/2$.
-- The Lissajous curves are also physically interesting, they describe the joint motion of orthogonal
uncoupled oscillators $(x(t),y(t),z(t))$ with different frequencies.

A {\bf prime knot} is not the knot sum of smaller knots. For example,
Square Knot and Granny Knot are not prime: each is a sum of two Trefoil Knots. 
There are 14 prime knots with the minimal number of crossings at most 7, see the documentation
(``About This Object'' or ATO) for ``V. Jones Braid List". 
 The 4th 7-crossings-knot, the prime knot $7_4$, is our default Lissajous space curve,
$(dd, ee, ff, gg) = (2, 3, 7, \pi/2)$. -- Two other alternating examples are: \hfill\break
\rightline{$(dd, ee, ff) = (2,5,13)$ resp. $=(4,3,23)$.\phantom{34534567}}

\vskip1pt
\hbox{
\vbox{\hsize=0.2\hsize \includegraphics[width=1.8 in]{LissajousKnot7_4.png}\vskip-10pt}
\hskip80pt
\vbox{\hsize=0.2\hsize \includegraphics[width=1.8 in]{Lissajous2_5_13.png}\vskip-10pt}
\hskip30pt
\vbox{\hsize=0.2\hsize \includegraphics[width=1.5 in]{Lissajous4_3_23.png}}
}
\goodbreak\eject
There are 249 prime knots with at most 10 minimal number of crossings. One can visualize
those via the Space Curves Menu entry: \break ``V. Jones Braid List''.
The notion of prime knot is important because Horst Schubert proved that the decomposition
of a knot as knot sum (= connected sum) of prime knots is unique. The knot invariants are
a good way to check whether a given knot is a prime knot. 
There is no  more elementary criterion to recognize a knot as prime.

There is an easy sufficient criterion that guarantees that the knot under consideration cannot
be drawn with fewer crossings. First we define {\it alternating} and {\it reduced alternating} knots:
if the thread of the knot passes alternatingly through
overcrossings and undercrossings then the knots is called {\bf alternating}.  For example, if
we twist a circle into a figure 8 we obtain an alternating trivial knot. In this case we observe an
easily recognizable property of the crossing in the knot diagram: if the crossing is removed the
knot diagram decomposes into two components. A crossing with this property is called an
{\it isthmus}. Clearly, one can always rotate one component of the knot diagram through 
180 degrees, untwist and thereby remove the isthmus to obtain a representation with one 
less crossings. An alternating knot without an isthmus is called a {\it reduced alternating knot}.

{\it Theorem} :
Reduced alternating knots cannot be represented with fewer crossings, they are always non-trivial. 


H.K.

\end{document}